In two posthumous papers, Jacobi proposed a sharp bound on the order of a system of n ordinary differential equations in n variables
ui(x)=0.
If ai,jis the order of the ith equation in the jth variable, he claims that the order of the system is bounded by
H=maxσÎSnΣi ai,σ(i).
We provide an English translation of the first of these texts, written in Latin. A French translation is available for the second one. The work of translation in English is in progress... One will also find above the original papers and the transcription of some related document from Jacobis Nachlaß.
It must be
noticed that these papers contain many interesting and forgotten results. The
first one contains, besides Jacobi’s bound, the first mention of what Ritt
called the "differential analog of Bézout’s theorem", that bounds the order of
the system by Σimaxj ai,j,
but it seems to have been "well known" at that time for Jacobi does just recall
it in the introduction. The second proves a bound on the minimal order one must
differentiate the equations in order to compute a normal form by elimination. He
also shows how to compute the orders of derivation of the system equations in
order to produce an auxiliary system, generically allowing to compute an
equation depending only of one chosen variable, that is a differential resolvent.
These two papers also show that the bound is reached if and only if an expression that Jacobi calls the truncated (determinans mancum ou determinans mutilatum) does not vanish.
Last, but
not least, Jacobi exposes a polynomial time algorithm to solve the following
problem: let (ai,j) be a square matrix n x n, find a permutation σ
such that Σ ai,σ(i) be maximal. This algorithm relies on the computation of a canon. Such an algorithm was only
rediscovered in 1955 par Kuhn, who called it "Hungarian method", having obtained
it from results Kőnig and Egerváry that relies on the notion of minimal cover. It is a well-known problem in mathematical
economy, under the name of "assignment problem". Working with canon or minimal covers turn to be equivalent.
You can also consult our translations published in a special issue of AAECC in 2009:
Carl Gustav Jacob Jacobi, "Looking
for the order of a system of arbitrary ordinary differential equations.
De investigando ordine systematis aequationum differentialium vulgarium
cujuscunque", translation from the Latin by F.O., AAECC 20, n° 1, 7-32, 2009. DOI Author's version
Carl Gustav Jacob Jacobi, "The
reduction to normal form of a non-normal system of differential
equations. De aequationum differentialium systemate non normali ad
formam normalem revocando", translation from the Latin by F.O., AAECC 20, n° 1, 33-64, 2009. DOI Author's version
or an unpublished paper devoted to the story of Jacobi's bound:
F. O., "Jacobi's Bound and Normal Forms Computations", Differential Algebra and Related Topics . Author's version ;